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The base case for the Euclidean algorithm is when the second argument of the pair is zero. At this point, the algorithm stops, and the first argument is returned as the greatest common divisor.

Conditions: The algorithm is in the process of replacing pairs using $\gcd(a,b)=\gcd(b,a\bmod b)$.; The second argument $b$ has reached zero.

The base case for the Euclidean algorithm is when the second argument of the pair is zero. At this point, the algorithm stops, and the first argument is returned as the greatest common divisor.

Conditions: The algorithm is in the process of replacing pairs using $\gcd(a,b)=\gcd(b,a\bmod b)$.; The second argument $b$ has reached zero.

For a nonnegative integer $n$, the greatest common divisor of $n$ and $0$ is $n$ itself. This serves as the base case for the Euclidean algorithm, terminating the recursion when the second argument becomes zero.

Conditions: $n$ is a nonnegative integer.; The pair is $(n, 0)$.

For a nonnegative integer $n$, the greatest common divisor of $n$ and $0$ is $n$ itself. This serves as the base case for the Euclidean algorithm, terminating the recursion when the second argument becomes zero.

Conditions: $n$ is a nonnegative integer.; The pair is $(n, 0)$.

The invariant property is that the set of common divisors of two numbers $a$ and $b$ is identical to the set of common divisors of $a$ and $b-a$ (or $b$ and $a \bmod b$). This ensures that the greatest common divisor (gcd) remains unchanged throughout the algorithm's steps.

Conditions: $a$ and $b$ are integers.; The operation is either subtraction ($b-a$) or modulo ($a \bmod b$).

The invariant property is that the set of common divisors of two numbers $a$ and $b$ is identical to the set of common divisors of $a$ and $b-a$ (or $b$ and $a \bmod b$). This ensures that the greatest common divisor (gcd) remains unchanged throughout the algorithm's steps.

Conditions: $a$ and $b$ are integers.; The operation is either subtraction ($b-a$) or modulo ($a \bmod b$).

The greatest common divisor of 10 and 15 is 5. This is determined by listing the common positive divisors of both numbers, which are 1 and 5, and selecting the largest one.

Conditions: The inputs are the positive integers 10 and 15.; The method used is listing all common positive divisors.

The greatest common divisor of 10 and 15 is 5. This is determined by listing the common positive divisors of both numbers, which are 1 and 5, and selecting the largest one.

Conditions: The inputs are the positive integers 10 and 15.; The method used is listing all common positive divisors.